# Direct Proportion: Definition, Formula, Symbol, Examples, FAQs

Home » Math Vocabulary » Direct Proportion: Definition, Formula, Symbol, Examples, FAQs

## What Is Direct Proportion in Math?

Direct proportion or direct variation is a type of proportion in which the ratio of two quantities stays constant.

Suppose a variable y varies directly with x, then it means that if x increases, y increases by the same factor. If x decreases, then there is a proportionate decrease in y also.

Direct Proportion Example:

The cost of the candy increases as the number of the same increases.

## Direct Proportion Symbol

The “directly proportional symbol” or “direct proportional symbol” is $\propto$.

We read x ∝ y as “x is directly proportional to y.”

It means that x is dependent on y.

We read y ∝ x as “y is directly proportional to x.”

It means that y is dependent on x.

## Direct Proportion Formula

If y is directly proportional to x, then the direct proportion formula or the direct proportion equation is given by y=kx, where k is a constant of proportionality.

## Constant of Proportionality

From the direct proportion formula, we have y = kx.

The fixed value $k = \frac{y}{x}$ is called the constant of proportionality and it represents the constant ratio between the two quantities that are in direct proportion. k can be any non-zero real number.

## Direct Proportion Graph

If you construct a graph of a direct proportion, it always comes out to be a straight line passing through the origin (0, 0).

The slope of this line is k.

• If k is negative, the line goes down from left to right.
• If k is positive, the line rises from left to right.

## Difference between Direct Proportion and Inverse Proportion

Direct proportion and inverse proportion are two kinds of proportional relationships in math describing the relation between two variables. Here are their key differences in the table below:

## How to Use Direct Proportion to Solve Problems

Let’s understand this with the help of an example.

Example: If 20 pens cost $25, what would be the cost of 100 pens? Here, the cost of pens is directly proportional to the number of pens. Note down the given values of x and y. Since the quantities are in direct proportion, their ratio is constant.$\frac{20}{25} = \frac{100}{?}$By cross multiplying, we get$20 \times ?= 100 \times 25$?$= \$125$

100 pens will cost $\$125$. ## Facts about Direct Proportion • Prior to$\propto$(symbol), a double colon (::) was used. • The proportionality symbol$(\propto)$was used by William Emerson (London, 1768) for the first time. • The link between the two variables is no longer a direct proportion if the proportionality ratio changes. ## Conclusion In this article, we learned about direct proportion, its graph, formula, equation, and examples. Let’s solve a few examples and practice problems to understand the concept better. ## Solved Examples of Direct Proportion 1. If 8 rooms are required for 24 guests, how many rooms would be required to accommodate 12 guests? Solution: Here,$\frac{24}{8} = \frac{w}{12}$Cross-multiply:$w \times 8 = 12 \times 24$w = 36 Therefore, for 12 guests, a total of 36 rooms will be required. 2. If 4 tasks take 8 hours for completion, how many tasks can be completed in 20 hours? Solution: Let’s write the constant ratios.$\frac{4}{8} = \frac{n}{20}n = 10$In 20 hours, 10 tasks can be completed. 3. What will 10 movie tickets cost if the price of 6 tickets is$\$36$?

Solution:

6 tickets cost $\$36$. Let 10 movie tickets cost$y.

$\frac{6}{36} = \frac{10}{6}$

$y \times 6 = 36 \times 10$